Jointly interpolating and deghosting seismic data

ABSTRACT

A technique includes representing actual measurements of a seismic wavefield as combinations of an upgoing component of the seismic wavefield and ghost operators. Interpolated and deghosted components of the seismic wavefield are jointly determined based at least in part on the actual measurements and the representation.

BACKGROUND

The invention generally relates to jointly interpolating and deghostingseismic data.

Seismic exploration involves surveying subterranean geologicalformations for hydrocarbon deposits. A survey typically involvesdeploying seismic source(s) and seismic sensors at predeterminedlocations. The sources generate seismic waves, which propagate into thegeological formations creating pressure changes and vibrations alongtheir way. Changes in elastic properties of the geological formationscatter the seismic waves, changing their direction of propagation andother properties. Part of the energy emitted by the sources reaches theseismic sensors. Some seismic sensors are sensitive to pressure changes(hydrophones), others to particle motion (e.g., geophones), andindustrial surveys may deploy only one type of sensors or both. Inresponse to the detected seismic events, the sensors generate electricalsignals to produce seismic data. Analysis of the seismic data can thenindicate the presence or absence of probable locations of hydrocarbondeposits.

Some surveys are known as “marine” surveys because they are conducted inmarine environments. However, “marine” surveys may be conducted not onlyin saltwater environments, but also in fresh and brackish waters. In onetype of marine survey, called a “towed-array” survey, an array ofseismic sensor-containing streamers and sources is towed behind a surveyvessel.

SUMMARY

In an embodiment of the invention, a technique includes representingactual measurements of a seismic wavefield as combinations of an upgoingcomponent of the seismic wavefield and ghost operators. Interpolated anddeghosted components of the seismic wavefield are jointly determinedbased at least in part on the actual measurements and therepresentation.

In another embodiment of the invention, a system includes systemincludes an interface and a processor. The interface receives actualmeasurements of a seismic wavefield, which are acquired by seismicsensors. The processor is adapted to represent the actual measurementsof the seismic wavefield as combinations of an upgoing component of theseismic wavefield and ghost operators; and the processor is adapted tojointly determine interpolated and deghosted components of the seismicwavefield based at least in part on the actual measurements and therepresentation.

In yet another embodiment of the invention, an article includesinstructions that are stored on a computer accessible storage mediumthat when executed by a processor-based system cause the processor-basedsystem to represent actual measurements of a seismic wavefield ascombinations of an upgoing component of the seismic wavefield and ghostoperators. The instructions when executed also cause the processor-basedsystem to jointly determine interpolated and deghosted components of theseismic wavefield based at least in part on the measurements and therepresentation.

Advantages and other features of the invention will become apparent fromthe following drawing, description and claims.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a schematic diagram of a marine acquisition system accordingto an embodiment of the invention.

FIGS. 2 and 3 are flow diagrams depicting techniques to jointlyinterpolate and deghost seismic data according to embodiments of theinvention.

FIG. 4 is a schematic diagram of a schematic data processing systemaccording to an embodiment of the invention.

DETAILED DESCRIPTION

FIG. 1 depicts an embodiment 10 of a marine seismic data acquisitionsystem in accordance with some embodiments of the invention. In thesystem 10, a survey vessel 20 tows one or more seismic streamers 30 (twoexemplary streamers 30 being depicted in FIG. 1) behind the vessel 20.The seismic streamers 30 may be several thousand meters long and maycontain various support cables (not shown), as well as wiring and/orcircuitry (not shown) that may be used to support communication alongthe streamers 30. In general, each streamer 30 includes a primary cableinto which is mounted seismic sensors that record seismic signals.

In accordance with embodiments of the invention, the seismic sensors aremulti-component seismic sensors 58, each of which is capable ofdetecting a pressure wavefield and at least one component of a particlemotion that is associated with acoustic signals that are proximate tothe multi-component seismic sensor 58. Examples of particle motionsinclude one or more components of a particle displacement, one or morecomponents (inline (x), crossline (y) and vertical (z) components (seeaxes 59, for example)) of a particle velocity and one or more componentsof a particle acceleration.

Depending on the particular embodiment of the invention, themulti-component seismic sensor 58 may include one or more hydrophones,geophones, particle displacement sensors, particle velocity sensors,accelerometers, pressure gradient sensors, or combinations thereof.

For example, in accordance with some embodiments of the invention, aparticular multi-component seismic sensor 58 may include a hydrophone 55for measuring pressure and three orthogonally-aligned accelerometers 50to measure three corresponding orthogonal components of particlevelocity and/or acceleration near the seismic sensor 58. It is notedthat the multi-component seismic sensor 58 may be implemented as asingle device (as depicted in FIG. 1) or may be implemented as aplurality of devices, depending on the particular embodiment of theinvention. A particular multi-component seismic sensor 58 may alsoinclude pressure gradient sensors 56, which constitute another type ofparticle motion sensors. Each pressure gradient sensor measures thechange in the pressure wavefield at a particular point with respect to aparticular direction. For example, one of the pressure gradient sensors56 may acquire seismic data indicative of, at a particular point, thepartial derivative of the pressure wavefield with respect to thecrossline direction, and another one of the pressure gradient sensorsmay acquire, at a particular point, seismic data indicative of thepressure data with respect to the inline direction.

The marine seismic data acquisition system 10 includes one or moreseismic sources 40 (one exemplary source 40 being depicted in FIG. 1),such as air guns and the like. In some embodiments of the invention, theseismic sources 40 may be coupled to, or towed by, the survey vessel 20.Alternatively, in other embodiments of the invention, the seismicsources 40 may operate independently of the survey vessel 20, in thatthe sources 40 may be coupled to other vessels or buoys, as just a fewexamples.

As the seismic streamers 30 are towed behind the survey vessel 20,acoustic signals 42 (an exemplary acoustic signal 42 being depicted inFIG. 1), often referred to as “shots,” are produced by the seismicsources 40 and are directed down through a water column 44 into strata62 and 68 beneath a water bottom surface 24. The acoustic signals 42 arereflected from the various subterranean geological formations, such asan exemplary formation 65 that is depicted in FIG. 1.

The incident acoustic signals 42 that are acquired by the sources 40produce corresponding reflected acoustic signals, or pressure waves 60,which are sensed by the multi-component seismic sensors 58. It is notedthat the pressure waves that are received and sensed by themulti-component seismic sensors 58 include “up going” pressure wavesthat propagate to the sensors 58 without reflection, as well as “downgoing” pressure waves that are produced by reflections of the pressurewaves 60 from an air-water boundary 31.

The multi-component seismic sensors 58 generate signals (digitalsignals, for example), called “traces,” which indicate the acquiredmeasurements of the pressure wavefield and particle motion. The tracesare recorded and may be at least partially processed by a signalprocessing unit 23 that is deployed on the survey vessel 20, inaccordance with some embodiments of the invention. For example, aparticular multi-component seismic sensor 58 may provide a trace, whichcorresponds to a measure of a pressure wavefield by its hydrophone 55;and the sensor 58 may provide one or more traces that correspond to oneor more components of particle motion, which are measured by itsaccelerometers 50.

The goal of the seismic acquisition is to build up an image of a surveyarea for purposes of identifying subterranean geological formations,such as the exemplary geological formation 65. Subsequent analysis ofthe representation may reveal probable locations of hydrocarbon depositsin subterranean geological formations. Depending on the particularembodiment of the invention, portions of the analysis of therepresentation may be performed on the seismic survey vessel 20, such asby the signal processing unit 23. In accordance with other embodimentsof the invention, the representation may be processed by a seismic dataprocessing system (such as an exemplary seismic data processing system320 that is depicted in FIG. 4 and is further described below) that maybe, for example, located on land or on the vessel 20. Thus, manyvariations are possible and are within the scope of the appended claims.

The down going pressure waves create an interference known as “ghost” inthe art. Depending on the incidence angle of the up going wavefield andthe depth of the streamer 30, the interference between the up going anddown going wavefields creates nulls, or notches, in the recordedspectrum. These notches may reduce the useful bandwidth of the spectrumand may limit the possibility of towing the streamers 30 in relativelydeep water (water greater than 20 meters (m), for example).

The technique of decomposing the recorded wavefield into up and downgoing components is often referred to as wavefield separation, or“deghosting.” The particle motion data that are provided by themulti-component seismic sensors 58 allows the recovery of “ghost” freedata, which means data that are indicative of the upgoing wavefield.

Deghosted and interpolated seismic data typically are essential for manyimportant seismic data processing tasks, such as imaging, multipleattenuation, time-lapse seismic processing, etc. In accordance withembodiments of the invention described herein, techniques are describedthat provide for concurrent, or joint, interpolation and deghosting ofthe acquired seismic data. More specifically, the seismic data areobtained by the regular or irregular sampling of the pressure andparticle motion data. As an example, these data may be sampled along oneor more streamers in the marine environment or may be sampled by seismicsensors located on the sea bed, as another example.

Techniques and systems are described herein that jointly interpolate anddeghost acquired seismic data. More specifically, based on themeasurements that are acquired by the multi-component sensors, anupgoing component of the pressure wavefield (herein called “p_(u)(x, Y;z_(s), f)”) component is determined at the seismic sensor locations, aswell as at locations other than the sensor locations, without firstinterpolating the acquired seismic data and then deghosting theinterpolated data (or vice versa).

The upgoing pressure wave component p_(u)(x, y; z_(s), f) at a temporalfrequency f and cable depth z_(s) may, in general, be modeled as acontinuous signal as the sum of J sinusoids that have complex amplitudes(called “A_(j)”), as set forth below:

$\begin{matrix}{{p_{u}\left( {x,{y;z_{s}},f} \right)} = {\sum\limits_{j = 1}^{J}{A_{j}{^{j\; 2{\pi {({{k_{x,j}x} + {k_{y,j}y} + {k_{z,j}z_{s}}})}}}.}}}} & {{Eq}.\mspace{14mu} 1}\end{matrix}$

In Eq. 1, “k_(x,j)” represents the inline wavenumber for index j;“k_(y,j)” represents the crossline wavenumber for index j; “z_(s)”represents the streamer tow depth; “f” represents the temporal frequencyof the sinusoids; and “c” represents the acoustic velocity in water.Additionally, “k_(z,j),” the wavenumber in the vertical, or depth,direction may be described as follows:

k _(z,j) =f/c√{square root over (1−c ²(k _(x,j) ² +k _(y,j) ²)/f²)}.  Eq. 2

Based on the representation of the upgoing pressure component p_(u)(x,y; z_(s), f) in Eq. 3, the pressure and particle motion measurements maybe represented as continuous signals described below:

$\begin{matrix}{{m^{P}\left( {x,{y;z_{s}},f} \right)} =} & {{Eq}.\mspace{14mu} 3} \\{{\sum\limits_{j = 1}^{P}{A_{j}{H\left( {k_{x,j},{k_{y,j};z_{s}},f} \right)}^{{j2\pi}{({{k_{x,j}x} + {k_{y,j}y} + {k_{z,j}z_{s}}})}}}},} & \;\end{matrix}$

where “m^(P)(x, y; z_(s), f)” represents a measurement vector, whichincludes the pressure and orthogonal components of the particle velocityin the inline, crossline and vertical coordinates, respectively. Thus,the measurements of the vector m^(P)(x, y; z_(s), f) are contiguous.

The measurement vector m^(P)(x, y; z_(s), f) may be described asfollows:

m ^(P)(x,y;z _(s) ,f)=[p ^(P)(x,y;z _(s) ,f)v _(x) ^(P)(x,y;z _(s) ,f)v_(y) ^(P)(x,y;z _(s) ,f)v _(z) ^(P)(x,y;z _(s) ,f)]^(T),  Eq. 4

where “H(k_(x),k_(y); z_(S), f)” represents a ghosting operator, whichis defined as follows:

$\begin{matrix}{{H\left( {k_{x},{k_{y};z_{s}},f} \right)} = {\begin{bmatrix}\begin{matrix}{\left( {1 + {\xi }^{{j4}\; {\pi k}_{z}z_{s}}} \right)\frac{{ck}_{x}}{f}} \\{\left( {1 + {\xi }^{j\; 4\; {\pi k}_{z}z_{s}}} \right)\frac{{ck}_{y}}{f}}\end{matrix} \\{\left( {1 + {\xi }^{{j4}\; {\pi k}_{z}z_{s}}} \right)\frac{{ck}_{z}}{f}\left( {1 - {\xi }^{{j4\pi k}_{z}z_{s}}} \right)}\end{bmatrix}^{T}.}} & {{Eq}.\mspace{14mu} 5}\end{matrix}$

In Eq. 5, “z_(s)” represents the streamer depth; and ζ represents thereflection coefficient of the sea surface.

Due to the relationships set forth in Eqs. 1 and 3, the A_(j) parametersmay be determined for purposes of jointly interpolating the acquiredseismic data and determining the upgoing pressure component p_(u)(x, y;z_(s), f).

More specifically, referring to FIG. 2, in accordance with someembodiments of the invention, a technique 120 to generate an upgoingcomponent of a seismic data wavefield, such as an upgoing pressurecomponent, includes representing (block 124) actual measurements of aseismic wavefield as combinations of an upgoing component of the seismicwavefield and ghost operators. Pursuant to the technique 120, theinterpolated and deghosted components of the seismic wavefield aredetermined based at least in part on the actual measurements and therepresentation, pursuant to block 128.

Eqs. 1 and 3 define the upgoing pressure component p_(u) (x, y; z_(s),f) and measurement vector m^(P)(x, y; z_(s), f) as a combination ofsinusoidal basis functions. However, it is noted that the componentp^(u)(x, y; z_(s), f) and the measurement vector m^(P)(x, y; z_(s), f)may be represented as a combination of other types of basis functions,in accordance with other embodiments of the invention.

If the sinusoids in Eq. 3 were not subject to the ghosting operators,then a matching pursuit technique could be used to identify theparameters of the sinusoids. The matching pursuit technique is generallydescribed in S. Mallat and Z. Zhang Mallat “Matching pursuits withtime-frequency dictionaries” IEEE Transactions on Signal Processing,vol. 41, no. 12, pp. 3397-3415 (1993). The matching pursuit algorithmmay be regarded as an iterative algorithm, which expands a particularsignal in terms of a linear combination of basis functions. As describedherein, the matching pursuit algorithm is generalized to the cases wherethe signal is represented as a linear combination of basis functionsthat are subject to some linear transformation, e.g., the ghostingoperation. This generalized technique described herein is referred to asthe Generalized Matching Pursuit (GMP) algorithm.

Referring to FIG. 3, in accordance with some embodiments of theinvention, a technique 150 may be used for purposes of determining thecoefficients of Eqs. 1 and 3. In this regard, the technique 150includes, pursuant to block 152, selecting a new basis function,applying the ghosting operator H(k_(x), k_(y); z_(s), f) to the newbasis function and adding the transformed basis function to an existingmeasurement function to form a new measurement function. In this regard,after the first basis function (which may be in the simplest form asingle sinusoidal function or even a constant) is added, a newexponential is added at every iteration to the set of basis functionsused, and the corresponding “ghosted” basis function is added to therepresentation; and then, an error, or residual, is determined based onthe actual seismic data that are acquired by the sensor measurements,pursuant to block 156.

The residual energy is then minimized for purposes of determining theA_(j) parameters for the new basis function. More specifically, adetermination is made (diamond 160) whether the residual energy has beenminimized with the current parameters for the new basis function. Ifnot, the parameters are adjusted and the residual energy is againdetermined, pursuant to block 156. Thus, a loop is formed for purposesof minimizing some metric of the residual energy until a minimum valueis determined, which permits the coefficients for the next basisfunction to be determined. Therefore, pursuant to diamond 168, ifanother basis function is to be added (based on a predetermined limit ofbasis functions, for example), the technique 150 continues with block152 to add the next basis function and calculate the correspondingparameters. Otherwise, if no more basis functions are to be added, theupgoing component of the seismic event is determined, pursuant to block174.

As a more specific example, the A_(j) parameters for the newest basisfunction may be determined by minimizing the energy of the residual.Therefore, if P−1 basis functions have been determined previously, therepresentation of the component p_(u)(x, y; z_(s), f) with the P−1sinusoids may be as follows:

$\begin{matrix}{{p_{u}\left( {x,{y;z},f} \right)} = {\sum\limits_{j = 1}^{P - 1}{A_{j}{^{{j2\pi}{({{k_{x,j}x} + {k_{y,j}y} + {k_{z,j}z_{s}}})}}.}}}} & {{Eq}.\mspace{14mu} 6}\end{matrix}$

The corresponding measurement function for the P−1 basis functions maybe obtained by applying the ghost operators to the basis functions:

$\begin{matrix}{{m^{P - 1}\left( {x,{y;z_{s}},f} \right)} \approx {\sum\limits_{j = 1}^{P - 1}{A_{j}{H\left( {k_{x,j},{k_{y,j};z_{s}},f} \right)}{^{{j2\pi}({{k_{x,j}y} + {k_{z,j}z_{s}}})}.}}}} & {{Eq}.\mspace{14mu} 7}\end{matrix}$

The residual in the approximation, called “r^(P−1)(x,y;z_(s),f)” may bedefined as follows:

$\begin{matrix}{{r^{P - 1}\left( {x,{y;z_{s}},f} \right)} =} & {{Eq}.\mspace{14mu} 8} \\{{m\left( {x,{y;z_{s}},f} \right)} - {\sum\limits_{j = 1}^{P - 1}{A_{j}{H\left( {k_{x,j},{k_{y,j};z_{s}},f} \right)}{^{{j2\pi}{({{k_{x,j}y} + {k_{z,j}z_{s}}})}}.}}}} & \;\end{matrix}$

If a new basis function “ĀPe^(j2π(k) ^(x,j) ^(y+k) ^(z,j) ^(z) ^(s) ⁾,”which has a corresponding coefficient called “ĀP” is added to theexisting representation of the upgoing wavefield, then the residual maybe rewritten as follows:

r _((Ā) _(P) _(, k) _(x,P) _(, k) _(y,P) ₎ ^(P)(x,y;z _(s) ,f)=r^(P−1)(x,y:z _(s) ,f)−Ā _(P) H( k _(x,P) , k _(y,P) ;z _(s) ,f)e^(j2π( k) ^(x,P) ^(x+ k) ^(y,P) ^(y+ k) ^(z,P) ^(z) ^(s) ⁾.  Eq. 9

It is noted that for Eq. 9, the parameters Ā_(P), k _(x,P), k _(y,P) forthe new basis function term are determined.

As a specific example, the parameters of the new basis function may befound by minimizing some metric of the residual, which is calculatedover inline and crossline sensor locations, as described below:

$\begin{matrix}{{\left( {k_{x,P},k_{y,P},A_{p}} \right) = {\arg {\min\limits_{({{\overset{\_}{A}}_{P},{\overset{\_}{k}}_{x,P},{\overset{\_}{k}}_{y,P}})}{M\left( {{\overset{\_}{A}}_{P},{\overset{\_}{k}}_{x,P},{{\overset{\_}{k}}_{y,P};z_{s}},f} \right)}}}},} & {{Eq}.\mspace{14mu} 10}\end{matrix}$

One such example metric may be described as follows:

$\begin{matrix}{{M\left( {{\overset{\_}{A}}_{P},{\overset{\_}{k}}_{x,P},{{\overset{\_}{k}}_{y,P};z_{s}},f} \right)} =} & {{Eq}.\mspace{14mu} 11} \\{\sum\limits_{m,n}{\left( {r_{({{\overset{\_}{A}}_{P},{\overset{\_}{k}}_{x,P},{\overset{\_}{k}}_{y,P}})}^{P}\left( {x,{y;z_{s}},f} \right)} \right)^{H}C^{- 1}}} & \; \\{{r_{({{\overset{\_}{A}}_{P},{\overset{\_}{k}}_{x,P},{\overset{\_}{k}}_{y,P}})}^{P}\left( {x,{y;z_{s}},f} \right)},} & \;\end{matrix}$

where “C” represents a four by four positive definite matrix; “x_(m)”represents the sensor locations in the inline direction; and “y_(n)”represents the sensor locations in the crossline direction.

Referring to FIG. 4, in accordance with some embodiments of theinvention, a seismic data processing system 320 may perform at leastsome of the techniques that are disclosed herein for purposes of jointlyinterpolating and deghosting seismic data. In accordance with someembodiments of the invention, the system 320 may include a processor350, such as one or more microprocessors and/or microcontrollers. Theprocessor 350 may be located on a streamer 30 (FIG. 1), located on thevessel 20 or located at a land-based processing facility (as examples),depending on the particular embodiment of the invention.

The processor 350 may be coupled to a communication interface 360 forpurposes of receiving seismic data that corresponds to pressure and/orparticle motion measurements. Thus, in accordance with embodiments ofthe invention described herein, the processor 350, when executinginstructions stored in a memory of the seismic data processing system320, may receive multi-component data that is acquired bymulti-component seismic sensors while in tow. It is noted that,depending on the particular embodiment of the invention, themulti-component data may be data that is directly received from themulti-component seismic sensor as the data is being acquired (for thecase in which the processor 350 is part of the survey system, such aspart of the vessel or streamer) or may be multi-component data that waspreviously acquired by the seismic sensors while in tow and stored andcommunicated to the processor 350, which may be in a land-basedfacility, for example.

As examples, the interface 360 may be a USB serial bus interface, anetwork interface, a removable media (such as a flash card, CD-ROM,etc.) interface or a magnetic storage interface (IDE or SCSI interfaces,as examples). Thus, the interface 360 may take on numerous forms,depending on the particular embodiment of the invention.

In accordance with some embodiments of the invention, the interface 360may be coupled to a memory 340 of the seismic data processing system 320and may store, for example, various input and/or output data setsinvolved with the techniques 120 and/or 150, as indicated by referencenumeral 348. The memory 340 may store program instructions 344, whichwhen executed by the processor 350, may cause the processor 350 toperform one or more of the techniques that are disclosed herein, such asthe techniques 120 and/or 150 and display results obtained via thetechnique(s) on a display (not shown in FIG. 4) of the system 320, inaccordance with some embodiments of the invention.

Other embodiments are within the scope of the appended claims. Forexample, in other embodiments of the invention, the seismic data may beacquired using another type of seismic acquisition platform, such as aset of ocean bottom cables, as a non-limiting example.

As additional examples of other embodiments of the invention, themeasurements that are obtained may be irregularly or regularly sampledwith respect to space and/or time. Additionally, the techniques that aredescribed herein may be used to determine a downgoing pressure orparticle motion component. Additionally, the techniques that aredescribed herein may be used with a subset of particle motionmeasurements (i.e., measurements in less than all three dimensions). Forexample, in accordance with some embodiments of the invention,interpolation may be performed in the cross-line direction and theseismic data may be deghosted when only pressure and the “vertical”component of the particle velocity are measured. As another example, theseismic data may be interpolated and deghosted when pressure, the“vertical” component of the particle velocity and the “cross-line”component of the particle velocity are used. Other variations arecontemplated and are within the scope of the appended claims.

As another variation, in accordance with some embodiments of theinvention, the measurement function may be represented as multiplesignals, where each signal is associated with a different frequencyband. In this regard, the signal for each frequency band may beseparately, or independently, interpolated. Additionally, differentspatial bandwidths may be used in the different frequency bands for therepresentation of the upgoing wavefield by the combined basis functions.It is noted that the different spatial bandwidths may be determined bythe speed of propagation of the signals.

While the present invention has been described with respect to a limitednumber of embodiments, those skilled in the art, having the benefit ofthis disclosure, will appreciate numerous modifications and variationstherefrom. It is intended that the appended claims cover all suchmodifications and variations as fall within the true spirit and scope ofthis present invention.

1. A method comprising: representing actual measurements of a seismicwavefield as combinations of an upgoing component of the seismicwavefield and ghost operators; and jointly determining interpolated anddeghosted components of the seismic wavefield based at least in part onthe actual measurements and the representation.
 2. The method of claim1, wherein the act of representing comprises representing the upgoingcomponent as a linear combination of basis functions.
 3. The method ofclaim 2, wherein the act of determining comprises determining parametersof the linear combination of the basis functions using a generalizedmatching pursuit technique.
 4. The method of claim 2, wherein the act ofdetermining comprises determining parameters of the linear combinationof basis functions in an iterative sequence of adding a basis functionto the existing linear combination of basis functions, and determiningcoefficients associated with the added basis function.
 5. The method ofclaim 4, wherein the act of determining the coefficients associated withthe added basis function comprises: applying the ghost operator to theadded basis function to generate a transformed basis function; addingthe transformed basis function to an existing representation of theactual measurements to generate a new representation of the actualmeasurements; and minimizing a residual between the actual measurementsand the new representation of the actual measurements.
 6. The method ofclaim 1, wherein the act of jointly determining comprises determining anupgoing component or a downgoing component of a pressure.
 7. The methodof claim 1 wherein the act of jointly determining comprises determiningan upgoing component or a downgoing component of a particle motion. 8.The method of claim 1, wherein the ghost operator comprises an operatorthat is a function of at least one of an acoustic velocity, a streamerdepth and a sea surface reflection coefficient.
 9. The method of claim1, wherein the actual measurements comprise a vector of pressure andparticle motion measurements.
 10. The method of claim 9, wherein theactual measurement vector comprises the three particle motion componentsin addition to the pressure component.
 11. The method of claim 9,wherein the actual measurement vector comprises any subset of pressureand particle motion components.
 12. The method of claim 1, wherein theactual measurements comprise measurements acquired by a spread of towedstreamers.
 13. The method of claim 1, wherein the actual measurementscomprise measurements acquired by a spread of over/under streamers or aset of ocean bottom cables.
 14. The method of claim 1, wherein theactual measurements comprise measurements acquired at regularly orirregularly spaced positions and/or times.
 15. The method of claim 1,wherein the actual measurements comprise measurements in athree-dimensional space.
 16. The method of claim 1, wherein the act ofdetermining comprises: representing the measurements as signalsassociated with a plurality of frequency bands; and determining thesignals independently.
 17. The method of claim 16, wherein the act ofdetermining further comprises: using different spatial bandwidths in thedifferent frequency bands for the representation of an upgoing wavefieldby combining basis functions.
 18. The method of claim 17, wherein thedifferent spatial bandwidths are based on the speed of propagation ofthe signals. 19.-33. (canceled)